maximal Subgroups in Composite Finite Groups

Abstract

The purpose of this paper is to present a method for translating the problem of finding all maximal subgroups of finite groups into questions concerning groups that are nearly simple. (A finite group is called nearly simple if it has only one minimal normal subgroup and that it nonabelian and simple.) In view of the recently announced classification of all finite simple groups this seems to be a useful reduction, though it must be acknowledged (see, e.g., Scott [6]) that there are still enormous obstacles on the way to understanding even just the maximal subgroups of the simple groups. Let G be a finite group and M a minimal normal subgroup of G. The maximal subgroups of G containing M are of course in bijective correspondence with the maximal subgroups of the smaller group G/M, and so we need not concern ourselves with those. If M is abelian, the maximal subgroups of G not containing M are precisely the complements of M in G; the number of conjugacy classes of these is 0 or the order of the first cohomology group ~‘(G/~, M). The case of A4 nona~lian and nonsimple is the principal part of this paper. If neither reduction is applicable, then all minimal normal subgroups of G are nonabelian simple groups: this is dealt with in the entirely straightforward penultimate section of the paper. These reductions are all “canonical” or “natural” in a sense which could perhaps be expressed in the language of categories and functors, but here we prefer to stay with older conventions. (In particular, we usually do not distinguish between a homomorphism and that obtained from it by restricting the codomain.) Nevertheless, the interested reader will observe that much of the strength of the results lies precisely in their canonical nature, implicit as it may remain in this exposition. While I do not know of any explicit statement of this reduction 114 002 l-8693/86 IE3.00

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